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Physics of Networks
Behrad TaghaviDepartment of Physics, University of Tehran, Tehran
WHAT WE ARE GOING TO TALK ABOUT?
What is called a Complex system ?
What is Social physics ?
An introduction to networks and graph theory
degree distribution in Networks
Resilience to the removal of nodes
Spreading processes on Networks
The small-world effect
Correlation and communities
COMPLEX SYSTEM
The most common and significant properties of complex systems are :
(i) The system is composed of a large number of interacting elements;
(ii)The interaction among the elements are nonlinear ;
(iii)Each element is unaware of the behavior of the system as a whole , it reacts only to locally available informations ;
(iv)The system is usually open and in a state far from equilibrium ; and
(v)Complex system have a history , their actual and future behavior depend upon this history and are particularly sensitive to it .
COMPLEX SYSTEM The toolbox available to study Complex systems drives from three main areas :
1. Nonlinear dynamics
2. Statistical Physics
3. Network Science
Some examples of Complex systems are :
1. Biological systems
2. Colonies and Communities of living things ( including human societies )
3. Financial systems ( Stock markets , ... )
and many other systems . Image Source: Wikipedia
COMPLEX SYSTEM The toolbox available to study Complex systems drives from three main areas :
1. Nonlinear dynamics
2. Statistical Physics
3. Network Science
Some examples of Complex systems are :
1. Biological systems
2. Colonies and Communities of living things ( including human societies )
3. Financial systems ( Stock markets , ... )
and many other systems . Image Source: Wikipedia
social physics is the study of social systems and social phenomena from the perspective of physics . Said another way, social physics is “the application of the concepts of physics in the social science” .
Mathematical tool we use to model a society is a Network .
SOCIAL PHYSICS
Image Source: PhD Comics
NETWORKSIn its simplest form , a network is a collection of points, or ‘nodes’ , joined by lines, or ‘edges’ .
As purely mathematical objects, networks ( or so-called graphs ) have been the subject of academic scrutiny since at least the 18th century . But they have taken a new practical role in recent years as a primary tool in the study of complex systems .
There are lots of complex systems that can be represented as a network ; Like : Internet, World Wide Web , Social networks of friendships between individuals , and many other systems.
Networks are also increasingly common in the study of Biological networks , epidemiology , computer viruses , genetics , human transportation and communication , and many other things .
“Leonhard Euler” is often credited with the first rigorous result in graph theory , with his solution of the famous ‘Königsberg bridge’ problem in 1765 . Kenneth Appel and Wolfgang Haken’s 1976 proof of the ‘four-color’ theorem is perhaps the best known recent achievement of graph theory .
NodeEdge
From looking at the network, it is evident that although the pattern of connections is not a regular one, neither is it completely random.
Image Source: The Physics of Networks, Mark Newman, Physics today 35, 2008.
DEGREE DISTRIBUTION IN A NETWORK
The degree of a node in a network is the number of edges connected to that node .
The degree distribution is one of the most basic quantitative properties of a network .
Networks in which node’s degree follow a poisson distribution ( ) are called ‘ER‘ ( Erdös-Rényi) networks.
Networks that follow a power law degree distribution ( ) are called ‘Scale free’ .
Image Source: The Physics of Networks, Mark Newman, Physics today 35, 2008.
RESILIENCE TO THE REMOVAL OF NODES
Question 1 : by some estimates , as many as 3% of routers world wide may be nonfunctional at any given time ; why internet network always stays connected though ?
Question 2 : networks like internet are fragile to removal of hubs ( removing only 3% of nodes is sufficient to disconnect the network ) . is this fragility always a negative point ?
Image Source: The Physics of Networks, Mark Newman, Physics today 35, 2008.
SPREADING PROCESSES ON NETWORKS
Image Source: The Physics of Networks, Mark Newman, Physics today 35, 2008.
SMALL-WORLD EFFECT
small world effect : most pairs of people , no matter how distant they may be , are connected by at least one and probably many short chains of acquaintances .
network of 80 nodes with rewiring probability of 0.3 and average path length of 3.663
Image Source: Wikipedia
CORRELATION AND COMMUNITIES
Image Source: The Physics of Networks, Mark Newman, Physics today 35, 2008.
THANKS
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